Effect of change in a on steady-state k in the Diamond model:
Let k* be described as Do
(1 – a) k*a,
where Do = 1 / (1+n)(1+g)(2+ r)
Differentiate k* with
respect to a:
dk* / da = d(Do (1 – a) k* a)
/ d a = Do
(1 – a) a k*a -1 – Do
k* a
dk*/da > 0 iff (1 – a) a k*a
-1 > k* a, or
(1 – a) a > k*
Now k* can be solved for
from the steady-state condition of this specific model:
k* = [Do (1 – a)]1/(1- a). Substituting in the condition above,
dk*/d a > 0 iff (1 – a) a > [Do (1 – a)]1/(1- a), or
[(1 – a)- a /(1- a) a]1- a > Do, or simplest of all,
a1- a /
(1 – a)a >
Do = 1 / [(1+n)(1+g)(2+r)]
This condition is not
necessarily true with the assumptions made in generating the model, since a and n and g and r can
take any positive on in some cases even negative values. However, consider an extreme case where n =
g = r = 0, so there is no growth of population or
efficiency and no time preference. Do
is ½ = 0.5 in that extreme case. Is
this greater than the left hand side expression? That depends on a. To see how, look at the table below,
calculated in Excel:
|
Values of
|
l
h s |
|
|
|
|
|
|
|
|
|
|
a = |
0.1 |
0.15 |
0.2 |
0.25 |
0.3 |
0.33 |
0.35 |
0.4 |
0.45 |
0.5 |
|
a exp(1-a) |
0.13 |
0.20 |
0.28 |
0.35 |
0.43 |
0.48 |
0.51 |
0.58 |
0.64 |
0.71 |
|
(1-a)^a |
0.99 |
0.98 |
0.96 |
0.93 |
0.90 |
0.88 |
0.86 |
0.82 |
0.76 |
0.71 |
|
ratio |
0.13 |
0.20 |
0.29 |
0.38 |
0.48 |
0.54 |
0.59 |
0.71 |
0.84 |
1.00 |
If we allow n and g to be 1
percent per year, or 28 percent per 25-year generation, and for 1 percent per
year time preference, Do becomes 1 / 3.75 = .27, smaller than any plausible
estimate of the left hand side, so it seems plausible to conclude that a rise
in the capital income share does raise steady state capital in our specific
Cobb-Douglas – logarithmic utility version of the Diamond model.
NC, Oct. 2004